Name______________________________________________ Calculus
Optimization Worksheet
HW #_______
Answer each of the following questions. Show all work for full credit. 1. A farmer wishes to fence a rectangular pasture adjacent to a river. The pasture must contain 180,000 square meters in order to provide enough grass for the herd. What dimensions would require the least amount of fencing if no fencing is needed along the river? 2. A Norman Window is constructed by adjoining a semicircle to the top of an ordinary rectangular window. Find the dimension of a Norman Window of maximum area if the total perimeter is 16 feet. 3. Find the coordinates of the point on the graph of y = x2 that is closest to (2, 0.5). 4. Find two positive numbers such that the sum of the first and twice the second is 100 and the product is a maximum. 5. A rancher has 200 feet of fencing with which to enclose two adjacent rectangular corrals. What dimensions should be used so that the enclosed area will be a maximum? 6. A rectangle is bounded by the x and y axes and the graph of y = − 12 x + 3 . What length and width should the rectangle have so that its area is a maximum? 7. A right triangle is formed in the first quadrant by the x and y axes and a line through the point (1,2). a. Write the length L of the hypotenuse as a function of x b. Use your graphing calculator to approximate x such that the length of the hypotenuse is a minimum. c. Find the vertices of the triangle such that the area is a minimum. 8. The range R of a projectile fired with an initial velocity v at an angle θ with the horizontal is v 2 sin 2θ R= g where g is the acceleration due to gravity. Find the angle θ such that the range is a maximum. 9. A man is in the desert 5 miles east of city A which is 5 miles north of city B along a road. If the man can drive 15 mph in the desert and 35 mph along the road, then at what location along the road should the man drive to so that the time to travel from where he is to city B is a minimum? 10. A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 12 cubic centimeters. Find the radius of the cylinder that produces the minimum surface area. 11. A light source is located over the center of a circular table of diameter 4 feet (see figure). Find the height h of the light source such that the illumination I at the perimeter of the table is maximum if k sin α I= s2 where s is the slant height, α is the angle at which the light strikes the table, and k is a constant.
h
s α
α 4
12. On a given day, the flow rate F, in cars per hour, on a congested roadway is v F= 22 + 0.02v 2 where v is the speed of the traffic in miles per hour. What speed will maximize the flow on the road? 13. An open box is to be made from a square piece of material, 24 inches on a side by cutting equal squares from the corners and turning up the sides. What size squares should be cut out to maximize the volume of the box? 14. A rectangle is bounded by the x-axis and the semicircle y = 25 − x 2 . What length and width should the rectangle have so that its area is a maximum? 15. Find the dimensions of the largest isosceles triangle that can be inscribed in a circle of radius 4 (see figure).
4
16. Determine the dimensions of a rectangular solid with square base and surface area of 150 square inches that maximizes its volume.
h
17. The sum of the perimeters of an equilateral triangle and a square is 10. Find the dimensions of the triangle and the square that produce a minimum total area. 18. A wooden beam has a rectangular cross section of height h and width w (see figure). The strength S of the beam is directly proportional to the width and the square of the height. What are the dimensions of the strongest beam that can be cut from a round log of diameter 24 inches? (Hint: S = kh 2 w , where k is the proportionality constant.)
w
24
h
19. Two factories are located at the coordinates (–x,0) and (x,0) with their power supply located at the point (0,h). Find y, between the power supply and the origin, such that the total amount of power line from the power supply to the factories is a minimum. 20. A man is in a boat 2 miles from the nearest point on the coast, denoted by A. He is to go to a point Q, 3 miles down the coast and 1 mile inland (see picture below). If he can row at 2 mph and walk at 4 mph, towards what point on the coast, denoted by P, should he row in order to reach Q in the least time?
Hint: Ttotal = Twater + Tland
2 miles 3–x A
x
P
D = rt 1 mile Q
3 miles
